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Let $D$ be a domain of $\mathbb{R}^d$. That is, $D$ is a connected open subset of $\mathbb{R}^d$. The first-order Sobolev space $W^{1,2}(D)$ on $D$ is defined by \begin{align*} W^{1,2}(D)=\{f \in L^2( …

The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates. Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of Neum …

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), j=1 …

Let $U$ be a bounded domain of $\mathbb{R}^d$, and write $m$ for the Lebesgue measure on $U$. For $k=1,2$, we denote by $H^k(U)$ the set of all locally $m$-integrable functions $u\colon U \to \mathbb …

I have a question about the traces of functions in $W^{1,2}$. Let $D$ be a connected open subset of $\mathbb{R}^d$.We denote $W^{1,2}(D)$ by \begin{align*} W^{1,2}(D)=\{f \in L^{2}(D,dx) \mid \parti …

I have a question about Sobolev spaces. In the following, we assume $d \ge 2$. Let $D$ be a domain of $\mathbb{R}^d$. That is, $D$ is a connected open subset of $\mathbb{R}^d$. Note that $D$ is not n …

I have a question about Sobolev spaces Let $U$ be a bounded Lipschitz domain of $\mathbb{R}^{d}$. $H^{1}(U)$ denotes the first order $L^2$-Sobolev space on $U$ with Neumann boundary condition. It i …

I have a question about a Dirichlet form. Let $D$ be a open subset of $\mathbb{R}^d$. Then, we can define $H^{1}(D)$ by \begin{equation*} H^{1}(D)=\{f \in L^{2}(D,dx):\frac{\partial f}{\partial x_i} …

I have a question about functions satisfying a condition. Let $D \subset \mathbb{R}^d$ be a Lipschitz domain. That is, for each $x \in \partial D$, there exists an open neighborhood $U$ of $x$ in $\ …

I have a question about weak derivatives. Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is th …